Related papers: Universal quadratic forms over multiquadratic fiel…

Let $E$ be an elliptic curve over the rational field $\mathbf{Q}$. Let $K$ be a quadratic extension over $\mathbf{Q}$. Let $\mathrm{ST}(E/K)$ dente the Shafarevich-Tate group of $E$ over $K$. We show that (under mild conditions on $E$) for…

We construct smooth rational real algebraic varieties of every dimension $\ge$ 4 which admit infinitely many pairwise non-isomorphic real forms.

Given two elliptic curves defined over a number field K, not both with j-invariant zero, we show that there are infinitely many $D\in K^\times$ with pairwise distinct image in $ K^\times/{K^\times}^2 $, such that the quadratic twist of both…

We prove that every sum of squares in the rational function field in two variables $K(X,Y)$ over a hereditarily pythagorean field $K$ is a sum of $8$ squares. More precisely, we show that the Pythagoras number of every finite extension of…

A subfield $K$ of $\bar{\mathbb{Q}}$ is $large$ if every smooth curve $C$ over $K$ with a rational point has infinitely many rational points. A subfield $K$ of $\bar{\mathbb{Q}}$ is $big$ if for every positive integer $n$, $K$ contains a…

In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic $p$-rational number fields of degree $2^{t}$ for any odd prime number $p$ and any integer $t \geq 1$. Using the criteria provided…

Let $h_{(m,k)}$ be the class number of $\mathbb{Q}(\sqrt{1-2m^k}).$ We prove that for any odd natural number $k,$ there exists $m_0$ such that $k \mid h_{(m,k)}$ for all odd $m > m_0.$ We also prove that for any odd $m \geq 3,$ $k \mid…

Let p and q be two positive primes. In this paper we obtain a complete characterization of quaternion division algebras H_K(p,q) over the composite K of n quadratic number fields. Also, in Section 6, we obtain a characterization of…

Let X be a smooth quartic surface not containing lines, defined over a number field K. We prove that there are only finitely many bitangents to X which are defined over K. This result can be interpreted as saying that a certain surface,…

In this paper, we characterize and study dynamical properties of cubic vector fields on the sphere $\mathbb{S}^2 = \{(x, y, z) \in \mathbb{R}^3 ~|~ x^2+y^2+z^2 = 1\}$. We start by classifying all degree three polynomial vector fields on…

Let $n\ge1$, $r\ge0$ and $s\ge0$ be integers satisfying $4+r+3 s\le3^{n+1}$. Given linear polynomials $f_{i}(x)=m_{i} x+n_{i}$ for $1 \le i \le r+s$, where the coefficients $m_{i} , n_{i}$ are positive integers satisfying certain…

Let $S = \{ \infty \} \cup S_f$ be a finite set of places of $\mathbb{Q}$. Using homogeneous dynamics, we establish two new quantitative and explicit results about integral quadratic forms in three or more variables: The first is a…

We prove an asymptotic formula for class numbers of totlally imaginary quartic number fields, ie for number fields of degree 4 over Q with only complex embeddings. After previous work for real quadratic fields (Sarnak) and complex cubic…

A polynomial transformation of the real plane $\Bbb R^2$ is a mapping $\Bbb R^2\to\Bbb R^2$ given by two polynomials of two variables. Such a transformation is called quadratic if the degrees of its polynomials are not greater than two. In…

For $\mathrm{O}(\mathrm{q},k)$, the orthogonal group over a field $k$ of characteristic 2 with respect to a quadratic form $\mathrm{q}$, we discuss the isomorphism classes of fixed points of involutions. When the quadratic space is either…

We consider a system of $R$ cubic forms in $n$ variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided $n\geq 25R$, we prove an asymptotic formula for the number of integer points in…

In this study, we determine all modular curves $X_0(N)$ that admit infinitely many cubic points.

We show that all $p$-adic quintic forms in at least $n>4562911$ variables have a non-trivial zero. We also derive new result concerning systems of cubic and quadratic forms.

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by $x$ over which there exist elliptic curves with good reduction everywhere and rational $j$-invariant is $\gg…

Let $f$ be a positive definite integral ternary quadratic form and let $r(k,f)$ be the number of representations of an integer $k$ by $f$. In this article we study the number of representations of squares by $f$. We say the genus of $f$,…